Quadratic Equations — Class 10 Mathematics (CBSE) explained

What you’ll learn

1. 1

   Lesson 1: From Situations to Quadratic Equations

   5 min
   • Turn the Prayer Hall Dimensions into an Equation
   • A Long History of Solving Quadratic Equations
   • Recognise the Standard Form
   • Represent John and Jivanti's Marbles
   • Connect Toy Count with Total Cost
   • Expand and Collect Terms Before Classifying
   • Squared Terms Can Cancel
2. 2

   Lesson 2: Recognise and Solve by Factorisation

   6 min
   • A Squared Term Remains After Rearrangement
   • An Apparently Cubic Equation Can Be Quadratic
   • Practise Classification and Mathematical Representation
   • A Root Makes the Equation True
   • Split the Middle Term and Set Each Factor to Zero
   • Factorisation Can Give a Negative Root
   • Repeated Factors Give Equal Roots
3. 3

   Lesson 3: Real Solutions and the Discriminant

   5 min
   • Choose the Root That Represents a Breadth
   • Use Factorisation in Equations and Daily-Life Cases
   • Use the Quadratic Formula for Real Roots
   • A Positive Discriminant Gives Distinct Real Roots
   • A Zero Discriminant Makes the Roots Equal
   • A Negative Discriminant Gives No Real Roots
   • Calculate the Discriminant Before Deciding
4. 4

   Lesson 4: Apply the Formula and Practise

   6 min
   • Use Geometry to Model the Pole Position
   • Test the Park Model for Real Roots
   • Apply the Formula and Locate the Pole
   • Find Equal Roots with a Fractional Constant
   • Determine Root Nature and Equal-Root Conditions
   • Decide Whether the Proposed Situation Is Possible
   • Recap: Formulate, Solve and Interpret

Key concepts

Historical context

Abraham bar Hiyya Ha Nasi gives complete solutions of different quadratic equations in Liber embadorum, published in Europe in 1145.

Definition and standard form

What matters is the degree after simplification and whether the equation can be written in standard form.

Classification through simplification

To decide whether an equation is quadratic, first simplify it.

The simplification warning

An equation that looks cubic can turn out to be quadratic.

Meaning of a root

The polynomial's zeroes and the equation's roots are the same. A quadratic polynomial has at most two zeroes, so a quadratic equation has at most two roots.

Quadratic formula

The quadratic formula gives another way to find roots.

Zero discriminant

The equation has two equal real roots.

Quadratic-formula application

The corresponding distance from gate A is twelve metres.

Exam focus

Board exam

CBSE assesses this chapter in the 2026-27 year-end board examination.

Syllabus

QUADRATIC EQUATIONS: 1. Standard form of a quadratic equation ax^2 + bx + c = 0, (a ≠ 0). 2. Solutions of quadratic equations (only real roots) by factorization, and by using quadratic formula. Relationship between discriminant and nature of roots. 3. Situational problems based on quadratic equations related to day-to-day activities to be incorporated. Competency: demonstrates strategies of finding roots and determining the nature of roots of a quadratic equation. Learning outcomes: solves quadratic equations using factorization and quadratic formula; determines the nature of roots using discriminant; formulates and solves problems based on real life context.

Exam note

Only real roots are in scope.

Exam note

The 2026-27 textbook states the quadratic formula at the start of 4.4 Nature of Roots (with the Sridharacharya history in 4.1) and uses it in examples; it has no completing-the-square derivation, so do not add one.

Paper pattern

Mathematics Standard: 80 marks, 3 hours, 38 questions. Section A: 18 MCQs + 2 assertion-reason (1 mark each); B: 5 VSA (2 marks); C: 6 SA (3 marks); D: 4 LA (5 marks); E: 3 case-study questions (4 marks). No calculators.

Question style

About 50% of board questions are competency-focused.

Practice in the app

89 board-pattern questions with marking schemes: 26 multiple choice (1 mark), 12 assertion–reason (1 mark), 19 very short answer (2 marks), 16 short answer (3 marks), 5 long answer (5 marks) and 11 case-based (4 marks).

Sample questions with marking points

Board-pattern questions from the AI Lessons practice set for this chapter. The marking points show what an examiner looks for; full model answers are in the app.

1. Very short answer2 marks

   While checking x = -1/2 in 6x² - x - 2 = 0, a student writes -x = -1/2. Correct this sign error and complete the verification.

   Marking points

   • Corrects -x to 1/2.0.5
   • Substitutes correctly into 6x² - x - 2.0.5
   • Simplifies to zero and concludes that -1/2 satisfies the equation.1

2. Short answer3 marks

   A student claims that only a positive value of k can make 2x² + kx + 3 = 0 have two equal real roots. Test this claim and give every possible value of k.

   Marking points

   • Obtain the discriminant k² - 24.1
   • Use the equal-root condition k² - 24 = 0.0.5
   • Obtain k² = 24.0.5
   • State both values k = 2√6 and k = -2√6.0.5
   • Reject the claim because both signs satisfy the zero-discriminant condition.0.5

3. Long answer5 marks

   A student claims, "An equation containing a squared term must be quadratic." Examine this claim using (i) (2x - 1)(x - 3) = (x + 5)(x - 1) and (ii) x(x + 1) + 8 = (x + 2)(x - 2). Expand and rearrange each equation, state the degree of the resulting polynomial equated to zero, and give a justified conclusion about the claim.

   Marking points

   • Correctly expands both sides of equation (i) as 2x² - 7x + 3 and x² + 4x - 5.1
   • Rearranges equation (i) to x² - 11x + 8 = 0 or an equivalent standard form.1
   • States that equation (i) is quadratic because the resulting polynomial has degree 2 and a nonzero squared-term coefficient.1
   • Expands both sides of equation (ii) and cancels the squared terms to obtain x + 12 = 0.1
   • States that equation (ii) has degree 1 and is not quadratic, rejecting the claim because the squared-term coefficient must remain nonzero.1

Frequently asked questions

More Class 10 Mathematics chapters

• Chapter 1Real Numbers4 lessons · 21 min→
• Chapter 2Polynomials4 lessons · 21 min→
• Chapter 3Pair of Linear Equations in Two Variables4 lessons · 24 min→

All CBSE Class 10 Mathematics guides →